Applied Statistics and Probability for Engineers 6th Edition solutions

Applied Statistics and Probability for Engineers 6
Problem 44E Let X be a geometric random variable with parameter p. Find the maximum likelihood estimator of p based on a random sample of size n .

Applied Statistics and Probability for Engineers 6
Problem 45E Consider the Poisson distribution with parameter ? Find the maximum likelihood estimator of ?, based on a random sample of size n.

Applied Statistics and Probability for Engineers 6
Let X be a random variable with the following probability distribution: \(f(x)=\left\{\begin{array}{ll} (\theta+1) x^{\theta}, & 0 \leq x \leq 1 \\ 0, & \text { otherwise } \end{array}\right. \) Find the maximum likelihood estimator of \(\theta\) based on a random sample

Applied Statistics and Probability for Engineers 6
Consider the shifted exponential distribution \(f(x)=\lambda e^{-\lambda(x-\theta)}, \quad x \geq \theta\) When \(\theta=0\), this density reduces to the usual exponential distribution. When \(\theta>0\), there is positive probability only to the right of \(\theta\). (a) Find t

Applied Statistics and Probability for Engineers 6
Consider the probability density function \(f(x)=\frac{1}{\theta^{2}} x e^{-x / \theta}\),\(\quad 0 \leq x<\infty, \quad 0<\theta<\infty\) Find the maximum likelihood estimator for \(\theta\).

Applied Statistics and Probability for Engineers 6
Let \(X_1,\ X_2,\ldots,\ X_n\) be uniformly distributed on the interval 0 to a. Show that the moment estimator of a is \(\hat{a}=2 \bar{X}\). Is this an unbiased estimator? Discuss the reasonableness of this estimator. Equation Transcription: Text Transcription: X_1, X_2,...,

Applied Statistics and Probability for Engineers 6
Consider the probability density function \(f(x)=c(1+\theta x), \quad-1 \leq x \leq 1\) (a) Find the value of the constant c. (b) What is the moment estimator for \(\theta\)? (c) Show that \(\hat{\theta}=3 \bar{X}\) is an unbiased estimator for \(\theta\). (d) Find the maximum lik

Applied Statistics and Probability for Engineers 6
The Rayleigh distribution has probability density function \(f(x)=\frac{x}{\theta}e^{-x{^2}}\), \(\quad x>0, \quad 0<\theta<\infty\) (a) It can be shown that \(E\left(X^{2}\right)=2 \theta\). Use this information to construct an unbiased estimator for \(\theta\). (b) Find the maximum likeliho

Applied Statistics and Probability for Engineers 6
Consider the Weibull distribution \(f(x)=\left\{\begin{array}{ll} \frac{\beta}{\delta}\left(\frac{x}{\delta}\right)^{\beta-1} e^{-\left(\frac{x}{\delta}\right)^{\beta}}, & 0<x \\ 0, & \text { otherwise } \end{array}\right. \) (a) Find the likelihood function based on a random sample

Applied Statistics and Probability for Engineers 6
Reconsider the oxide thickness data in Exercise 7-35 and suppose that it is reasonable to assume that oxide thickness is normally distributed. (a) Compute the maximum likelihood estimates of \(\mu\) and \(\sigma^{2}\). (b) Graph the likelihood function in the vicinity of \(\hat{\mu}\) and \(\hat{\s

Applied Statistics and Probability for Engineers 6
Suppose that X is a normal random variable with unknown mean \(\mu\) and known variance \(\sigma^{2}\). The prior distribution for \(\mu\) is a normal distribution with mean \(\mu_{0}\) and variance \(\sigma_{0}^{2}\). Show that the Bayes estimator for \(\mu\) becomes the maximum likelihood estimator

Applied Statistics and Probability for Engineers 6
Suppose that X is a normal random variable with unknown mean and known variance \(\sigma 2\). The prior distribution for \(\mu\) is a uniform distribution defined over the interval [a, b]. (a) Find the posterior distribution for \(\mu\). (b) Find the Bayes estimator for \(\mu\).

Applied Statistics and Probability for Engineers 6
Suppose that X is a Poisson random variable with parameter \(\lambda\). Let the prior distribution for \(\lambda\) be a gamma distribution with parameters m+1 \((m+1) / \lambda_{0}\) . (a) Find the posterior distribution for \(\lambda\). (b) Find the Bayes estimator for \(\lambda\).

Applied Statistics and Probability for Engineers 6
Suppose that X is a normal random variable with unknown mean and known variance \(\sigma^{2}=9\). The prior distribution for \(\mu\) is normal with \(\mu_{0}=4\) and \(\sigma_{0}^{2}=1\). A random sample of \(n=25\) observations is taken, and the sample mean is \(\bar{x}=4.85\). (a) Find the Bayes es

Applied Statistics and Probability for Engineers 6
The weight of boxes of candy is a normal random variable with mean \(\mu\) and variance 1/10 pound. The prior distribution for \(\mu\) is normal with mean 5.03 pound and variance 1/25 pound. A random sample of 10 boxes gives a sample mean of \(\bar{x}=5.05\) pounds. (a) Find the Bayes estimate of \(\

Applied Statistics and Probability for Engineers 6
The time between failures of a machine has an exponential distribution with parameter \(\lambda\). Suppose that the prior distribution for \(\lambda\) is exponential with mean 100 hours. Two machines are observed, and the average time between failures is \(\bar{x}=1125\) hours. (a) Find the Bayes est

Applied Statistics and Probability for Engineers 6
Transistors have a life that is exponentially distributed with parameter \(\lambda\). A random sample of n transistors is taken. What is the joint probability density function of the sample?

Applied Statistics and Probability for Engineers 6
Problem 62SE Suppose that a random variable is normally distributed with mean ?and variance ?2 , and we draw a random sample of five observations from this distribution. What is the joint probability density function of the sample?

Applied Statistics and Probability for Engineers 6
Suppose that X is uniformly distributed on the interval from 0 to 1. Consider a random sample of size 4 from X . What is the joint probability density function of the sample?

Applied Statistics and Probability for Engineers 6
A procurement specialist has purchased 25 resistors from vendor 1 and 30 resistors from vendor 2. Let \(X_{1,1},\ X_{1,2},\ldots,\ X_{1,25}\) represent the vendor 1 observed resistances, which are assumed to be normally and independently distributed with mean 100 ohms and standard deviation 1.5 ohms.

Applied Statistics and Probability for Engineers 6
A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12. Find the probability that the sample mean is in the interval \(47 \leq X \leq 53\). Is the assumption of normality important? Why? Equation Transcription: Text T

Applied Statistics and Probability for Engineers 6
Problem 66SE A random sample of n = 9 structural elements is tested for compressive strength. We know that the true mean compressive strength = 5500 psi and the standard deviation is ? = 100 psi. Find the probability that the sample mean compressive strength exceeds 4985 psi.

Applied Statistics and Probability for Engineers 6
A normal population has a known mean 50 and known variance \(\sigma^{2}=2\). A random sample of \(n=16\) is selected from this population, and the sample mean is \(\bar{x}=52\). How unusual is this result? Equation Transcription: Text Transcription: sigma^2=2 n=16 x bar=52

Applied Statistics and Probability for Engineers 6
Problem 68SE A random sample of size n = 16 is taken from a normal population with = 40 and ?2 = 5 Find the probability that the sample mean is less than or equal to 37.

Applied Statistics and Probability for Engineers 6
A manufacturer of semiconductor devices takes a random sample of 100 chips and tests them, classifying each chip as defective or nondefective. Let \(X_{i}=0\) if the chip is nondefective and \(X_{i}=1\) if the chip is defective. The sample fraction defective is \(\hat{P}=\frac{X_{

Applied Statistics and Probability for Engineers 6
Let X be a random variable with mean \(\mu\) and variance \(\sigma^{2}\). Given two independent random samples of sizes n1 and n2 , with sample means \(\bar{X}_{1}\) and \(\bar{X}_{2}\), show that \(\bar{X}=a \bar{X}_{1}+(1-a) \bar{X}_{2}, \quad 0<a<1\) is an unbiased est

Applied Statistics and Probability for Engineers 6
A random variable x has probability density function \(f(x)=\frac{1}{2 \theta^{3}} x^{2} e^{-x / \theta}, \quad 0<x<\infty, \quad 0<\theta<\infty\) Find the maximum likelihood estimator for \(\theta\). Equation Transcription: Text Transcription: f(x)=1 ove

Applied Statistics and Probability for Engineers 6
Let \(f(x)=\theta x^{\theta-1}\), \(0<\theta<\infty\) and \(0<x<1\). Show that \(\hat{\Theta}=-n /\left(\1n \prod_{i=1}^{n} X_{i}\right)\) is the maximum likelihood estimator for \(\theta\). Equation Transcription: Text Transcription: f(x)=theta x^theta-1 0<theta<infinity 0<x<1

Applied Statistics and Probability for Engineers 6
Let \(f(x)=(1 / \theta) x^{(1-\theta) / \theta}\), \(0<x<1\), and \(0<\theta<\infty\). Show that \(\hat{\Theta}=-(1 / n) \sum_{i=1}^{n} 1 n\left(X_{i}\right)\) is the maximum likelihood estimator for \(\theta\) and that \(\hat{\Theta}\) is an unbiased estimator for q. Equation Tran

Applied Statistics and Probability for Engineers 6
You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is \(\mu\), which is unknown. You are interested in estimating the area of the square, which is\(\mu^{2}\). Because \(\mu\) is unknown, you measure it n times, obtaining observations \(X_1,\

Applied Statistics and Probability for Engineers 6
Problem 75SE An electric utility has placed special meters on 10 houses in a subdivision that measures the energy consumed (demand) at each hour of the day. The company is interested in the energy demand at one specific hourthe hour at which the system experiences the peak consumption. The data from

Applied Statistics and Probability for Engineers 6
A lot consists of N transistors, and of these, \(M(M \leq N)\) are defective. We randomly select two transistors without replacement from this lot and determine whether they are defective or nondefective. The random variable \(X_{i}=\left\{\begin{array}{ll} 1, & \text { if the } i

Applied Statistics and Probability for Engineers 6
When the sample standard deviation is based on a random sample of size n from a normal population, it can be shown that S is a biased estimator for \(\sigma\). Specifically, \(E(S)=\sigma \sqrt{2 /(n-1)} \Gamma(n / 2) / \Gamma[(n-1) / 2]\) (a) Use this result to obtain an unbiased est

Applied Statistics and Probability for Engineers 6
Consistent Estimator. Another way to measure the closeness of an estimator \(\hat{\Theta}\) to the parameter \(\theta\) is in terms of consistency. If \(\hat{\Theta}_{n}\) is an estimator of \(\theta\) based on a random sample of n observations, \(\hat{\Theta}_{n}\) is consistent for \(\theta\) if

Applied Statistics and Probability for Engineers 6
Order Statistics. Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size \(n\) from \(X\), a random variable having distribution function \(F(x)\). Rank the elements in order of increasing numerical magnitude, resulting in \(X_{(1)}, X_{(2)}, \ldots, X_{(n)}\), where \(X_{(1)}\) is the smalle

Applied Statistics and Probability for Engineers 6
Let \(X_1,\ X_2,\ldots,\ X_n\) be a random sample of a continuous random variable with cumulative distribution function \(F(x)\). Find \(E\left[F\left(X_{(n)}\right)\right]\) and \(E\left[F\left(X_{(1)}\right)\right]\)

Applied Statistics and Probability for Engineers 6
Let X be a random variable with mean ? and variance \(\sigma^2\), and let \(X_{1}, X_{2}, \ldots, X_{n}\), be a random sample of size n from X. Show that the statistic \(V=k \sum_{i-1}^{n-1}\left(X_{i+1}-X_{i}\right)^{2}\) is an unbiased estimator for \(\sigma^2\) for an appropriate choice for the co

Applied Statistics and Probability for Engineers 6
When the population has a normal distribution, the estimator \(\hat{\sigma}=\operatorname{median}\left(\left|X_{1}-\bar{X}\right|,\left|X_{2}-\bar{X}\right|, \ldots,\left|X_{n}-\bar{X}\right|\right) / 0.6745\) is sometimes used to estimate the population standard deviation. This estimator is

Applied Statistics and Probability for Engineers 6
Censored Data. A common problem in industry is life testing of components and systems. In this problem, we assume that lifetime has an exponential distribution with parameter \(\lambda\), so \(\hat{\mu}=1 / \hat{\lambda}=\bar{X}\) is an unbiased estimate of \(\mu\). When n components are tested until